Shape invariant modelling of trial based fMRI data

Introduction Most previous component analyses of fMRI data have avoided working directly with the tri-linear structure, but resorted to bi-linear models such as ICA, PCA, and NMF. Multi-linear decomposition can exploit consistency over trials and contrary to bilinear decomposition render unique representations without additional constraints. However, they can degenerate if data does not comply with the given multi-linear structure, e.g., due to shape variations. Materials and Methods The tri-linear CANDECOMP/PARAFAC (CP) of space-time-trial neuro-imaging data decompose the data into spatial and temporal signatures while accommodating trial-to-trial amplitude variability [1]. The model is however unable to accommodate trial variability such as the inevitable inter-trial delay and shape variations of typical neuroimaging experiments. Delay modelling has recently been treated using the ShiftCP model in [3], however, multiple delays which can occur for instance due to the cardiac or respiratory cycles or shape differences in activity between trials cannot be modelled within the ShiftCP representation. To accommodate such trial variability we propose the convolutive CP model (ConvCP), as illustrated in figure 1. The ConvCP model was updated alternating between (a, b, c) according to the following negative log-posterior (disregarding constants and with a and b normalised componentwise),